Suppose that $\varphi (t), \psi (t), w(t)$ are continuous functions on $[a,b]$ such that $w(t)>0$. If the inequality
$$\varphi (t)\leq \psi (t) + \int_a^t w(s)\varphi (s)ds$$
holds on $[a,b]$, prove that
$$\varphi (t)\leq \psi (t) + \int_a^t w(s)\psi (s) \exp \left\{\int_s^t w(u)du\right\}ds$$.
This was a homework problem from my ODE course that used Birkhoff Rota's ODE book. But I didn't know how this inequality is related to the theory of ODE and I have failed in proving it, please helps.
Let be $t\in\left[a,b\right]$ . Let be$$\chi\left(t\right)=\intop_{a}^{t}\omega\left(s\right)\varphi\left(s\right)ds.$$ I let you check that $\chi$ is $\mathcal{C}^{1}$ and that $\chi'\left(t\right)=\omega\left(t\right)\varphi\left(t\right)$ . Now by the hypothesis, we have$$\chi'\left(t\right)\leq\omega\left(t\right)\left(\psi\left(t\right)+\chi\left(t\right)\right)=\omega\left(t\right)\psi\left(t\right)+\omega\left(t\right)\chi\left(t\right)$$ i.e.$$\chi'\left(t\right)-\omega\left(t\right)\chi\left(t\right)\leq\omega\left(t\right)\psi\left(t\right).$$ We multiply the both sides of the last inequality by the same positive function and get$$\left[\chi'\left(t\right)-\omega\left(t\right)\chi\left(t\right)\right]e^{-\intop_{a}^{t}\omega\left(s\right)ds}\leq\omega\left(t\right)\psi\left(t\right)e^{-\intop_{a}^{t}\omega\left(s\right)ds}$$ i.e.$$\frac{d}{dt}\left[\chi\left(t\right)e^{-\intop_{a}^{t}\omega\left(s\right)ds}\right]\leq\omega\left(t\right)\psi\left(t\right)e^{-\intop_{a}^{t}\omega\left(s\right)ds}.$$ Now we integrate over $\left[a,t\right]$ and obtain the new inequality$$\chi\left(t\right)e^{-\intop_{a}^{t}\omega\left(s\right)ds}\leq\intop_{a}^{t}\omega\left(s\right)\psi\left(s\right)e^{-\intop_{a}^{s}\omega\left(s'\right)ds'}ds$$ (I let you check where is the term evaluated at $s=a$ on the left side after the integration). Multiplying the both sides by the same positive function, we get$$\chi\left(t\right)\leq e^{\intop_{a}^{t}\omega\left(s\right)ds}\intop_{a}^{t}\omega\left(s\right)\psi\left(s\right)e^{-\intop_{a}^{s}\omega\left(s'\right)ds'}ds=\intop_{a}^{t}\omega\left(s\right)\psi\left(s\right)e^{\intop_{s}^{t}\omega\left(s'\right)ds'}ds$$ and using the fact that $\chi\left(t\right)\geq\varphi\left(t\right)-\psi\left(t\right)$ , we obtain$$\varphi\left(t\right)-\psi\left(t\right)\leq\intop_{a}^{t}\omega\left(s\right)\psi\left(s\right)e^{\intop_{s}^{t}\omega\left(s'\right)ds'}ds$$ as claimed.